Short on time? This page is the recap. For the derivations, Figures 1 and 2 and the worked examples, use the full study guide. This version of the topic is for the calculus-based course.
Recap
- A capacitor is two separated conductors holding +Q and −Q. Its net charge is zero.
- Capacitance C = Q/ΔV, measured in farads (1 F = 1 C/V). It depends on shape, size and the gap material only.
- Between large parallel plates the two sheet fields, each σ/(2ε₀), add to σ/ε₀. Outside they cancel. The field is uniform except for fringing near the edges.
- Any shape: assume ±Q → Gauss’s law for E → integrate for ΔV → C = Q/ΔV.
- A charged particle between plates has constant acceleration qE/m: projectile-style motion, parabolic path.
- The work done to separate the charge is stored as potential energy.
- ε₀ = 8.85 × 10⁻¹² C²/(N·m²); assume an air gap (κ = 1) unless told otherwise.
Key relationships
| Situation | Result |
|---|---|
| Definition | C = Q/ΔV |
| Parallel plates, field | E = σ/ε₀ = Q/(ε₀A); ΔV = Ed |
| Parallel plates, capacitance | C = ε₀A/d (κε₀A/d with a dielectric) |
| Concentric spheres, radii a < b | C = 4πε₀ab/(b − a); b → ∞ gives 4πε₀a |
| Coaxial cylinders, radii a < b, length L | C = 2πε₀L/ln(b/a) |
| Stored energy | U = ∫(q/C)dq = Q²/(2C) = ½QΔV = ½C(ΔV)² |
| Particle between plates | a = qE/m, constant; across the gap y = ½at² |
Assumptions behind the results
- Plate separation is much smaller than plate size, so edge effects are ignored.
- Conductors are in electrostatic equilibrium; charge sits on the facing surfaces.
- Gravity is negligible for electrons, protons and ions in typical fields.
Mistakes to avoid
- σ/(2ε₀) between two plates. Two plates give σ/ε₀.
- Thinking C depends on Q or ΔV. It does not.
- Not asking what is fixed. Isolated: Q fixed. Battery connected: ΔV fixed.
- U = QΔV. Charging stores only ½QΔV.
- ε₀A/d for spheres or cylinders. Use the matching formula unless the gap is thin.
Quick self-check
- Plates of area 0.010 m² are 0.50 mm apart in air. What is C? (1.77 × 10⁻¹⁰ F, or 177 pF)
- An isolated capacitor’s plate gap is doubled. What happens to E and to U? (E unchanged; U doubles)
- How much energy does a 2.0 μF capacitor store at 50 V? (2.5 × 10⁻³ J)
Next: practice questions.